Luận văn Đánh giá lớp phép biến hình á bảo giác lên hình vành khăn bị cắt theo các cung tròn đối xứng quay

Lu(m van Th(lcsj Toan h(Jc- TntclngThu(1n 28 Chuang5 cA C DANH GIA LOP HAM G Trangchu'angnay,chungtoidanhgiacaed£;liu'cjngd~ctru'ngchami~n chuffnclingnhu'modulicuacaelOphamG. Vi~cdanhgia bankfnh Q(g), 9 E G, dongm9tvai tro quailtrQngtrangvi~cdanhgiacaed£;li lu'cjngkhac,VI the'chungWi b~tdftuvoi danhgianay. 5.1 Danh gia bankinh Q(g) DinhIy 5.1. VcJicaegiGthiefvaky hi£luiJ ehLtdng2, V9 E G, tae6 K (1+ S~~)r'<Q(g)</i-*, trongd6 dangthuetraixdy ra khi B = Bovag(w)= alwIK-1w,w E 1 1 B, lal =1vadangthuephdixdyrakhiB =Bovag(w)=blwlK-W,wE B, Ibl= 1. (5.1) Chungminh.Ap d\lngb6d~4.1chaPBHKABG ngu'cjcf =g-l, 9 E G, taco 2 2 32 > 81 (Q~g)r+ps(R~9)r 2 > 81 (Q~g))K+ps (doR(g)<1). Tli day,suyfa c~ndu'oicuaQ(g) (5.2) Q(g)>C2.~psrIf =(1+ S~~)rIf. Khi B = Bovag(w)=alwIK-1w,la!= 1thld~ngthucKayfa. M~Hkhac, ne'uf.1> 1,apd\lng[13,dinh191]choPBHKABG9 E G tanh~ndllc;1c 2 7r12 >7rQ2(g){lK . Suy fa 1 Q(g)<{l-K. Tli day,nhd [14,c6ngthuc2.5],ta co d~ngthucKayfa khi B =Bo -k-1 ~ vag(w)=blwl w, Ibl= 1, wEB. Ne'u{l= 1thldaubgiac~ntfenIa hi~nnhienvad~ngthuckh6ng th~Kayfa. 0 H~qua5.1.VI 8(B) 7rq2,cg,ndztcJicua Q(g) trong (5.1) dzt(fcvief dztcJid(lng ( P8) -if Q(g)>qK 1- -:; (gE G), (5.3) suy ra Q(g)>qK. (5.4) Deingthac(5.3)ho(ic(5.4)xdyra {:} B =Bovag(w)=alwIK-1w,w E B, lal= 1. 5.2 Danhgiac~ndtioibankinhR(g) K Vi R(g) > Q(g),\/g E G, tli dinh195.1,taco R(g) > (1+S~~))-2. M~tkhac,taconco dinh19sail: DinhIy 5.2.V6i caegidthiefvakyhi?u iJ chztdng2, \/9 E G va8>0, taco caedanhgia: R(g)>( 1- 8(B) ) 1f 82 j R(g)> ( p8 2 ) If . 82- 81{lKI (5.5) (5.6) 29 Cluingminh.Thea(5.2),voi 8>a ( 1 ) 1< ( 1 ) 1< -* 82>81 Q(g) +p8 R(g) >R (81+p8). Suy fa --k 82R <- 81+p8 Vi 81+ p8 =82- 8(B), taco (5.5). M~itkhac,tli (5.7)va (5.1),taco 2 ( 1 ) 1< 82>81 (1'*)K +p8 R(g) . Tli day suy fa (5.6). 5.3 CaedanhgiakhaeehoQ(g),R(g)va Ig(w)1 Blob Iy 5.3. VJi caegia thittva ky hi£1u(J chZlcJng2, 'l/gE G, taco 1 -x Q(g) < J-L ) , RK(p,Iwl,q)< Ig(w)1< RK (p, 1~I,q)- RK (p, 1;I,q-K Q(g) < f-l , (qK<)RK(p,d,q)<R(g)<RK (p,~,q)- RK (p,~,q) RK(p,d,q)RK(p,~,q)<Q(g). (5.7) D (5.8) (5.9) (5.10) Cluingminh.Ap d\lngbfftd~ngthucphai cua (4.20)cha PBHKABG w = j(z), z E A, hamngu'Qccuaz = g(w),wEB, voi 9 E G, taco Iwl < t voi t = T [v,Ig(w)I*",q] . Do do, theodinhnghlacuahai hamsf{ph\!T(p, r, 8) va R(p,t,8), E G va Vw E B, taco 1 Ig(w)IK=R(p,t,q) 30 varheatinhdondi~u(3.11)cuahamR(p,t,s),taco R(p,t,q)> R(p, Iwl,q). Tli do suy ra c~nduoi cua Ig(w) I trong(5.8). M~t khac, ta nh~nduQctli ba'td~ngthlic trai cua (4.20), \:/gE G va \:/wE B, Iw I >;v6it'=T [p,C~~i I) k ,q] . Tli do tu'dngt1/nhu'tren,\:/gE G va \:/wE B, taco ( ) -k ( . ) Q(g) I q Ig(w)1 =R(p,t, q)>R p,~, q , Ke'thcjpvoi (5.1),suyra cacc~ntrencua Ig(w)i trong(5.8). Tu'ongt1/,nhoba'td~ngthlic(4.22),tacoth~chira cacba'td~ngthlic (5.9). f)anhgiac~ndu'oi(5.10)d6i voi Q(g)duQcsuyra tr1/ctie'pt11(5.9). D H~qua5.2. Tit (3.24),tanh(mdu(lccacdanhgiaddngilln. vfii cacgill thiefva ky hi~uiJ chuang2, \:/gE G, taco 4-*lwIK < Ig(w)1< 4{fQ(g) C~I) K <4*p-k C~I)K, K K ( ) K K 1 ( ) K 4-P dK < R(g) <4PQ(g) ~ <4P jL-K ~ ' 4-~(CJ )K < Q(g)< 1c R(g) , 4_2f,' (q:r<Q(g). (5.11) (5.12) (5.13) (5.14) Tit (3.12)va (3.16),tathflyrlingcach~slf chiphl:lthuQcvaaK vap trang(5.11)-(5.14)la totnh{{t. Chzly 1. Truonghcjpcacthanhph~nbieno-jthoaihoathanhp di~m roi r(;lc,hi~nnhiendanhgia (5.11)vftncondungne'utathaycacda'u< bdi <. Nhu'v~y, bAngcach thactri~nlien t\lCham z =9(w) t(;lip di~m 31 biendfinell,tathffy(5.11)v~ncondungchoPBHKABG d6ixungquay p lfinz =g(w) mi~nnhilienB nQitie'ptfonghlnhvanhkhanq < Iwl < 1 leDhlnhvanhkhanQ < Izi < 1. Chuj 2. (5.14)coth~s~chon(5.4)khiq -+ 0, C -+ 0 vdi di~uki~n d " q=canstva - =canst. c H~qua5.3. TruiJnghC;pC1, C2 va cac CJj lan ll1c;tla cacdl1(Jngtran Iwl = Qt,Iwl =1vacacnhatcdttrenduiJngtranIwl= R', tacocac danhgiasau ~ , 1 Q'K <Q <QK, (5.15) ,1 RK (p,R',Q')<R < K (Q~, ') . R p,R" Q Trang(5.15)dlingthactraixayra{::}B =Bo, g(w)=alwIK-1w,w E 1 1 B, lal =1vadlingthacphaixayra{::}B =Bo,g(w)=blwlK-w, w E B, Ibl=1. (5.16) Changminh.Th~tv~y,khi do q = M1 = Q', C = d = R', J.t= J" S(B) = 7r(1- Q'2), 81= 7rQ'2,8= O. Do do,apd1;lngdinh195.1,tanh~ndu'Qc(5.15),clingvdi di~uki~n xayfa ding thuc. Tu (5.9)va (5.15),nh~ndu'Qc(5.16). D H~qua5.4. Ktt hC;p(5.2)vai (5.9),tatiml(Ii c(lndualcilaQ(g) co thi sdch(Jn(5.1)nhusau ( 1 ) 1< ( 1 ) 1< 82 > 81 Q(g) +p8 R(g) ( 1 ) 1< ( K ( q ) -k) -k > 81 Q(g) +p8 R p,~,q J.t . Suy ra Q(g)> ( 82- p8.R2(p,~,q) J.t~ ) -~ 81 . 32 H~qua5.5. Tit (5.9)va (5.10),tanh(mdLt(le,nhiJ (3.12)va (3.17),cae danhgia sauday ddi vai Ided{jh{jiI¥ ctlaR(g), ~i~ivaQ(g)lrongeae tntiJngh(lpgifJi h(ln K1f2 1- R(g) <1- RK(p,d,O) ~ K[l- R(p,d,O)] ~ 1 82p n p(l-d) khi d -+1,tllela R(g) -+ 1khi d -+ 1. 1- Q(9) < 1- RK ( CJ 0)R(g) P, c' ~ !{ [1- R (p,~,O)] ~ K1f28 C 2p In p(l-~) khi ~ -; I, tlic fa ~i:i -; 1khi ~-; 1. 1-Q(g) < 1-RK(p,d,0)RK(p,~,0) ~ K [(1- R(p,d,0))+(1- R (p,~,0))] !{1f2 K 1f2 ~ 8 + 8 2pInp(l-d) 2pInp(l-~) khi d -+1 va ~-+ 1. e FJanh giG(5.19)n6i r2ingQ(g) ddntdi 1ntu d -+ 1va:l. -+ 1.c 5.4 Danhgiag6cmdj3(g) (5.17) (5.18) (5.19) R5 rangtaluanco0 <{3(g)<21f,9 E G, tuynhientamu6ncodanhgia p t6thdntrongnhfi'ngtru'onghejpnaodo. Mu6nv~ytadungphu'dngphap dQd~li-di~ntichhaycongQiIa dQdaiQtctri doAhlforsvaBeurling[1] d~xu'ongnam1950,giupgiiHquye'tnhi~ubaitoant6iu'utrongPBHBG. Md rQngphu'dngphapdo choPBHKABG, taco b6d~sau: B6 d~5.1. Trongm(ltphdngz ehohlnhehilnh~t D = {z= x + iyl 0 <x < a, 0 <y < b}. 33 Gia sa ham so'W = j(z) th1!chi~nmQtPBHKABG hlnh chTlnh(ltD ZenmQtta giac GongH cila m(itphdngW saDcho cac dlnh 0, a, a + ib va ib cila D ztmZuffttu(jngang V(ji cac dlnh WI, W2,W3va W4cila H. GQi r ZahQcac cungr trongH noi cc;mhWIW2wJi c(;mhW3W4cila H. GiGsa co hamdQdo p = p(w) >0 lien tf:lCtrong H saD cho 0 < Ip(r) = lp,dW,<00, V"{ E r va 0<SetH)=JJ HP2dudv<00, W = u +iv. lJ(it lp = inf lp(r ). fEr Khi do, ta co 1 a 2 Sp(H) >K blpo Ddng thac (j (5.20) co thl xay ra. (5.20) W4 D ,Dx WI W3 ib a + ib 0 x a W2 Hinh5.1:PBHKABG hlnhchunh?tD JentugiaccongH. Changminh. *Tru'dngh<jpK =1 D~t 5x=Dn{zl~z=x}varx=j(5x), O<x<a. Theagiiithie't,taco a Sp(H) =JJ Dp21f'(zWdxdy= J dx1.p21f'(z)12Idyl. 0 34 Theoba'td~ngthucSchwarzI, tanh~ndu'QC\Ix E (0,a) L pV(zWldyll, Idyl>(L pl!'(z)lIdyl)2, vadofoxIdyl=b>0Denco l p21f'(zWldvl > ~ (l plf'(zJ[[dvl)2 . Do do,d€ y1xE f, taco a 2 a 2 SetH) > i J (l plf'(z)lIdYI)dx= i J (1. pldwl) dx0 0 a 12 J a2 > y;lp dx =y;lp. 0 *Tru'onghQpK > 1. Xet T/=h(w) la PBHBGtugiacH leDhlnhchii'nh~t D' = {'TJ= S+it I 0<s <a', 0<t <b'} saochocacdlnhWI, W2,W3va W4cuaH l~nhiQttu'ongumgvdi cac dlnh0, a',a'+ib'vaib'cuaD'. Apdvngchungminhtrenchoanhx~ngu'Qch-I, taco a' Sp(H)> bll~. M~tkhac,anhx~h0 f la PBHKABGhlnhchii'nh~tD leDhlnhchii' nh~t D' Denco a' 1 a ->-- b' - K b. IBilt dAngthucco d~ng !<g(X))'dx!<h(X))'dx;' Ug(x)h(x)dx) , trongdogiii thie'tg(x),h(x) lient\lcteendo~n[Xl,X2]vadAngthucxiiyfakhivachikhig(x)=Ch(x), X E [Xl,X2],C = canst. 35 Tli doco(5.20). f)~ngthuc(j (5,20)co th€ xayfa, ch~ngh~nkhi H tIlingvoi D', ( ) 1 '- a K a' Th" " kh' d' 1 al2 1 a(b')2 'b' P W = vab = b" ~t v~y, 1 0 K bP = K b =a = Sp(H). D B6d~5.2.Trangm(itphangz ehomiin E = {zl rl < Izi < r2, 'PI < argz < 'P2}. Gia sithamsa W= j(z) thl;tehi<fnmQtPBHKABG mi€n E Zen~Qt ta giac GongH cila m(itphangw sao chocaedlnhZl = rl ei'P2, Z2 = ei'Pl, Z3 = r2ei<plva Z4=r2ei'P2cila E tanZuqtuangringvdi caedlnh WI, 'W2,W3vaW4cilaH. GQir fahQcaecung"( trong H m5'iC(mhWIW2 vdi qmhW3W4cilaH. Vai caeky hi<fup, lp("(),"(E f, lp,Sp(H) nhutrongb6d€ 5.1,taco Sp(H) > ~'P2-r:ll~, (5.21)In- rI Dangthric(j (5,21)co thi xay ra. Z4 ~r2!r) A'f)1/'?2 Z 3 01"\ 1'9, Z2 /67 W4 W, H '"'II{>WI W2 i In r2 rl ~-!.pi +i In r2 rl ~6x 0 -x ~ -\PI Hinh5,2:PBHKABG mi~nE Jentti'giaccongH. 36 Changminh. Quaphepbie"nd6i z =x+iff=rp2+i In~ = (rp2 - rp)+i In~, z =reirp, rl rl mi~nE sebie"nthanhhlnhchITnh~t D ={z=x+iYI 0 <x< 'P2 - 'p" 0 <y <In~:} , d' , d? h - 0 - - .1 r2 ,,- .1 ,r2tfong 0 cac m Zl = , Z2= rp2-rpl,Z3= rp2-rpl+1,n - va Z4= 1,n- rl rl IftnhtCjttu'ongling vdi Zl, Z2,Z3va Z4,va cac cling Arp,6x,1rptu'ongling vdi nhaunhu'tfonghlnh 5.2. Tli b6 d~5.1, suy fa (5.21). D Dinh Iy 5.4. VcJieaegiGthilt va ky hi<fuiJ ehu(Jng2, V9 E G, wEB va 0 < q < M1 < C < d < m2 < M2 = 1,taco eaebatdangthue: 21r- p{3I 2 m2 + p{3I 2~ + p{3I 2 m2 <Ks (B)1 nM RnM Ind- Po'In - 1 In- 1 In- Q Q R 21r- p{3 In2m2+ p{3 In2~ + p{3 In2m2 <K2S (B). In (4~qCd) M1 In (4~~) M1 In (4~~) d Po trongdo (5.22) (5.23) fi dudv .Spa(B) = 2 2' W=U+'lV.BU +V Dangthac(j (5.22)co thl xay ra. Changminh.f)~t Arp=A n {zI argz =rp} vavdim6ij (j =0,...,p- 1),d~t Qlj = {z Q2j = {z Q3j = {z Q<lzl<l, -a+(2j+1)7r <argz<a+(2j+1)7r } , p p Q <Izi <R, ex+ (2j - 1)7r<argz <-0: +(2j + 1)7r} , P P R < Izi <1, ex+ (2j - 1)7r<argz < -a + (2j+1)7r} . P P 37 1Hinh5.3:PBHKABG mi~nchuffnA lenmi~nB lingvdi (p=2). PBHKABG ngu<;1cf=g-1 bie'nmi~nA leu mi~nB, trongdo Qlj bie'nthanhH1j comQtc~nhtrenC1vamQtc~nhtrenc2, Q2jbie'nthanh H2j co mQtqmhn~mtrenC1va mQtc~nhn~mtrenO"j,Q3jbie'nthanh H3j co ffiQtqlllh namtren(J"jva mQtc~nhnamtrenc2. Giasufk lftn1u'<;1tla hQcaccling1ktrongHkj (k =1,2,3) lftnhi<;1t n6i C1vdi C2,n6i C1voi O"j,n6i O"jvoi C2va gia sa co p = p(w) > 0 lien t\1CtrongB saocho 0 < lp(1k)=1 pldwl<00, 'V1k E fk (k =1,2,3)"fk va 0 <Sp(Hkj)=jf p2dudv<00 (k= 1,2,3),w =u+iv.Hkj £)~t lkp= inf Ip(1k) (k = 1,2,3). "fkEfk D~dangtha"y rang f1::) U f(A<p)vdi f1 = U A<p '\pEfi -£1+(2j+1)~::;<p::;a+(2j+l)~ Tit do, ap dvngb6 d.; 5.2 cho mi.;nQ'j, yiJi p(w) = I~I'wEB, co 1 20: 2 !{---yllp < Sp(H1j).In- Q vdi m2 hp = inf lp(11)=J - 1 1 1 1dwl = InMm2,"fiEfi W 1 fvh 38 tucHi 2'if- p{3In2m2 <pKSp(HI),1 M1In- Q (5.24) VI 2po;=2'if - p{3. 1 Tu'dngttf,d6ivdimi~nQ2j,vdip(w)= Iwl'wEB, co f2 ~ U f(A'P)voi r2= U 1'1" A.pEfz (X+(2j-l)~:::;'P:::;-(X+(2j+l)~ ( 'if ) 2 -0; 1 P Z2 <Sp(H2j).- R 2p- !( In - Q vdi c . J 1 C l2p = Inf lp(12)= _ I I idwi =In M 'bEG w 1 Ml tile IiI p{3In2~ <pKSp(H2j). R M1-In- Q (5.25) Tu'dngtif, d6i vdi mii\n Q3j, vdi p(w) =I~I'WEB, co f3 ~ U f(A'P) vdi r3= U 1'1" A<pEf3 (X+(2j-l)~:::;'P:::;-(X+(2j+l)~ ( 'if ) 2 -0; 1 P Z2 <Sp(H3j).- 1 3p- !( In - R vdi mz l3p = inf lp(13)=J -I l lldwi=Inmd2,I'3Ef3 w d 39 tucIa PP 2m2 ~In d <pKSp(H3j).In- R Tli (5.24),(5.25)va(5.26),VOlchu9 pK(Sp(H1j)+Sp(H2j)+Sp(H3j))=KSpo(B) (5.26) tanh~nduqc(5.22). Danhgia(5.23)duqcSuyfatu(5.22)vah~qua5.2. Ne'uB = Bo,tucml = M1 = q,C= d = r, m2= M2= 1 vane'u 1 1 1 Z = g(w)= IwIK-w, tucIzl = IwlK hayIwl= IzIK,q = QK, r = RK, thl K t ". (522) - 21r- PP12 1+ PP 12r + PP 12 1ve ral . n - - n - - n - In~ q In R q In~ r Q Q R - 21r- PP12~ PP 12 RK PP 1 2~ - 1 n QK + R n QK + 1 n RKIn - In - In - Q Q R 1 R 1 = I{2(21r- pp)In Q + I{2p{3InQ + K2p{3lnR 1 1 1 = 21rK2 In Q = 21rK In QK = 21r!{Inq' K ? . 5 J dudv ve phal( .22) = K 2 2U +v q<lwl<l 27f 1 27f 1 - K J J ~d:2de= K J deJ ~~=27rKln~. 0 q 0 q V~y,taco dAngthuc(j (5.22). 0 Chl1j 3. CacbfttdAngthuc(5.22)va(5.23)v~ncondungkhim2<M1 ho~cC <M1 ho~cm2<d ne'uta IftnIuqtd~tIn :~=0ho~cIn;1 =0 ho~cIn:2 =0,d6ngnghIaVOlvi<$ctakhongd~9de'ns1;id6ngg6pcua di~nHchcaet~pconcuaB khongthoagia thie'tdinh195.4. 40 H~qua5.6. Vlii caegidthiefvaky hi~uiJ chu(Jflg2, V9 E G, wEB va 0 < q < M1< C< d < m2< M2= 1,kef h(Jp(5.23)vlii biftdang 27r thachiln nhien0 <{3(g)< -, taco danhgia cho{3(g),VgE G: p ( [ ( 2 )] } K 2I 1 I 415c27r n- n d 27r max 0,- 1 - q 2 q <{3(g)<-.p In ~ pM1 Chuy 4.Khim2 = 1,M1=q,taxet 27r [ J(2In.! In (4~s..) ] - 1 - q qd P In2rn2 =M1 27r [ K2InlIn (4js.. )] - 1 - q qd P In21q - 27r [ 1- K21n(4~q'J) ] p Inl.q ( 2 )K2 In 411s.. D~tC = I 1 qd , trongtru'onghQpd=canst,choq---+0,C ---+0n- q saGcho~=canst,hlC ---+O. V~y27r>{3(g)> 27r(1- C), tilec~n q p p du'dicua{3(g)trongh~qua5.6Ia ffiQtdaubgias~c,it rachotru'onghQp daneUe 41